Events
Department of Mathematics and Statistics
Texas Tech University
Interacting particle models are often employed to gain understanding of the emergence of macroscopic phenomena from microscopic laws of nature. These individual-based models capture fine details, including randomness and discreteness of individuals, that are not considered in continuum models such as partial differential equations (PDE) and integral-differential equations. The challenge is how to simultaneously retain key information in microscopic models as well as efficiency and robustness of macroscopic models. In this talk, I will discuss how this challenge can be overcome by elucidating the probabilistic connections between particle models and PDE. These connections also explain how stochastic partial differential equations (SPDE) arise naturally under a suitable choice of level of detail in modeling complex systems. I will also present some novel scaling limits including SPDE on graphs and coupled SPDE. These SPDE not only interpolate between particle models and PDE, but also quantify the source and the order of magnitude of stochasticity. Scaling limit theorems and new duality formulas are obtained for these SPDE, which connect phenomena across scales and offer insights about the genealogies and the time-asymptotic properties of the underlying population dynamics. Watch online Tuesday the 10th at 3 PM via this Zoom link -- passcode 427144
 | Tuesday
Nov. 10
3:30 PM
MATH 017
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| Real-Algebraic Geometry
Affine Schemes
David Weinberg
Department of Mathematics and Statistics, Texas Tech University
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With travel becoming more frequent across the world, it is important to understand how spatial dynamics impact the spread of diseases. Human movement plays a key part on how a disease can be distributed as it enables a pathogen to invade a new environment and helps the persistence of a disease in locations that would otherwise be isolated. In this project, we explore how spatial heterogeneity combines with mobility network structure to influence vector-borne disease dynamics by using cellphone data from Namibia. In addition, we derived an approximation for the domain reproduction number for a n-patch SIR Ross-MacDonald model using a Laurent series expansion. Lastly, we will analyze the sensitivity equations with respect to the domain reproduction number to determine which parameters should be targeted for intervention strategies. PDF available
After establishing clearly a notion of global orbit category (of which there are several variants in the literature), we describe a class of topological stacks locally modeled on action ∞-groupoids with singularities via cohesive shape. In passing to the smooth case to obtain orbifolds as certain differentiable stacks, we describe V-folds as a formulation of étale ∞-groupoids internal to a differentially cohesive ∞-topos, which are also the groundwork for studying e.g. G-structures in this setting.In 1988, Avramov, Kustin, and Miller gave a complete classification of the possible Tor-algebra structures arising from the minimal free resolution of a quotient ring $R/I$ with projective dimension 3. Absent from this classification was a complete description of which Tor-algebra structures actually arise as the Tor-algebra of some quotient $R/I$ with some prescribed homological data. This problem of realizability has remained open, with recent progress by Christensen, Veliche, and Weyman, giving stronger restrictions on the set of Tor-algebra classes that may occur. In this talk, we will explore how the process of "trimming" an ideal can be used to preserve Tor-algebra class while altering homological data. We will see how an explicit algebra structure on a certain resolution of these ideals descends to multiplication on the Tor-algebra to explain why trimming an ideal often yields another ideal with the same Tor-algebra class. Finally, we discuss applications to the realizability problem for Tor-algebras.
Join Zoom Meeting
https://zoom.us/j/91512227044?pwd=NlRhRnFOYm1ROENGNFYrQy9TZGNXZz09
Meeting ID: 915 1222 7044
Passcode: 623310
This project has developed a class of bound preserving and energy dissipative schemes for the porous medium equation. The schemes are based on a positivity preserving approach and a perturbation technique, and are shown to be uniquely solvable, bound preserving, and in the first-order case, also energy dissipative. We have also conducted ample numerical experiments to validate the theoretical results and demonstrate the new schemes' effectivenessFollowing through on the promises for Cartan geometry in the first two talks, we formulate Haefliger stacks and G-structures in an elastic ∞-topos, the latter as a special case of the principal ∞-bundle constructions available in any ∞-topos where now the existence of the infinitesimal disk bundle is key. By introducing V-folds with singularities, in the sense of singular (elastic) cohesion, we promote étale ∞-stacks in differential cohesion to higher orbifolds in singular cohesion so as to obtain geometrically structured higher orbifolds, extending the intrinsic étale cohomology of étale ∞-stacks to tangentially twisted proper orbifold cohomology.In this study we suggest a portfolio selection framework based on time series of stock log-returns, option-implied information, and multivariate non-Gaussian processes. We empirically assess a multivariate extension of the normal tempered stable (NTS) model and of the generalized hyperbolic (GH) one by implementing an estimation method that simultaneously calibrates the multivariate time series of log-returns and, for each margin, the univariate observed one-month implied volatility smile. To extract option-implied information, the connection between the historical measure P and the risk-neutral measure Q, needed to price options, is provided by the multivariate Esscher transform. The method is applied to fit a 50-dimensional series of stock returns, to evaluate widely-known portfolio risk measures and to perform a forward-looking portfolio selection analysis. The proposed models are able to produce asymmetries, heavy tails, both linear and non-linear dependence and, to calibrate them, there is no need of liquid multivariate derivative quotes.
Joint work with Gian Luca Tassinari, University of Bologna